Abstract
The account of causal regularities in the influential INUS theory of causation has been refined in the recent developments of the regularity approach to causation and of the Boolean methods for inference of deterministic causal structures. A key element in the refinement is to strengthen the minimality or non-redundancy condition in the original INUS account. In this paper, we argue that the Boolean framework warrants a further strengthening of the minimality condition. We motivate our stronger condition by showing, first, that a rationale for strengthening the original minimality condition in the INUS theory is also applicable to our proposal to go further, and second, that the new element of the stronger condition is a counterpart to a well-established minimality condition for probabilistic causal models. We also compare the various minimality conditions in terms of the difference-making criteria they imply and argue for the criterion implied by our condition. Finally, we show that putative counterexamples to our proposal can be addressed in the same way that the Boolean theorists defend the current minimality conditions in their framework.
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Notes
The sense of sufficiency and necessity in question is regularity-theoretic and will be made precise in Sect. 2.
Abuse of notation: given any binary factor X, we write ‘X’ to express the presence of the factor and ‘~X’ to express its absence.
Mackie uses this qualifier to accommodate special cases where the condition expressed by a single literal is already sufficient or is part of a condition that is both sufficient and necessary.
Another minimality condition, called ‘structural minimality,’ is proposed by Baumgartner and Falk (2020). Since it is not a constraint on the causal regularity for a single factor, which is our concern here, we will leave it aside. Everything we say is compatible with adding the structural minimality in the end as a constraint on the conjunction of multiple causal regularities.
We thank a referee for drawing our attention to this point.
Pearl’s (2000) Theorem 1.4.1 entails the direction of ‘if’. The direction of ‘only if’ is trivial, for in the causal graph, every exogenous variable is parentless and is a non-descendant of any other exogenous variable.
Basically, if NR3 holds, then for any X∈W, no Y∈PAX is independent of X conditional on PAX\{Y}, and so no arrow can be taken away without violating the Markov condition; conversely, if NR3 fails, then for some X∈W, some Y∈PAX is independent of X conditional on PAX\{Y}, and hence the arrow from Y to X can be deleted without violating the Markov condition.
There are many variants of controlled experiments in practice, but we submit that whenever factors that are believed or hypothesized to be directly relevant are not controlled for, it is due to practical limitations rather than the belief that there is no need to control for them.
We thank a referee for noting this important point.
We thank a referee for suggesting this way of phrasing the worry.
A referee raised the question: what exactly is the evidence against the causal relevance of E1 to E2 in the data presented in Table 2? In particular, which difference pairs would tell against it? In our view, it is not the presence of any difference pair that serves as evidence against a hypothesis of causal relevance, but the absence of a certain type of difference pair. From the NR2 theorist’s point of view, there is no such evidence in Example 2, because the desired type of difference pair is not absent regarding the causal relevance of E1 to E2, whereas from the NR3 theorist’s point of view, the desired type is absent, which is the reason to drop the hypothesis (perhaps only temporarily, unless Table 2 is already nomologically complete). Whether such absence is best construed as evidence of absence (of causal relevance) or as absence of evidence (for causal relevance), we do not attempt to answer in this paper. On the latter construal, we agree with the referee that the elimination of the hypothesis in question may be said to be “pragmatic” (as opposed to “evidential”) in nature. We need not decide here which construal is right or better, because either way, we can make our point that the dialectical situation is parallel to that between the NR1 theorist and the interventionist.
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Acknowledgements
We are very grateful to Michael Baumgartner for extremely helpful discussions and to Clark Glymour for insightful comments on earlier drafts. This paper has also been improved by feedback from Lorenzo Casini, Kam-ching Leo Cheung, Zhiheng Tang, Kai-yee Wong, James Woodward, Tung-Ying Wu, Yujian Zheng, Lei Zhong, and two referees. This research was supported in part by the Research Grants Council of Hong Kong under the General Research Fund 13602720. Kun Zhang was also partially supported by the NIH under Contract R01HL159805, by the NSF-Convergence Accelerator Track-D award 2134901, by a grant from Apple Inc., a grant from KDDI Research Inc, and generous gifts from Salesforce Inc., Microsoft Research, and Amazon Research.
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Zhang, J., Zhang, K. A New Minimality Condition for Boolean Accounts of Causal Regularities. Erkenn (2023). https://doi.org/10.1007/s10670-023-00685-4
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DOI: https://doi.org/10.1007/s10670-023-00685-4